1. ## Pairing problem

Each of the 75 children in a line was assigned one of the integers from 1 through 75 by counting off in order. Then, standing in the same order, the children counted off in the opposite direction, so that the child who was assigned the number 75 the first time was assigned the number 1 the second time. Which of the following is a pair of numbers assigned to the same child?

How can I show and prove that it has to be 47 and 29?

2. ## Re: Pairing problem

Hi, mjoshua.

It might help to think about what happens when you add the numbers together that a student received.

Does this help get things on the right track?

Good luck!

3. ## Re: Pairing problem

It does not, I'm sorry.. Anyone else?

4. ## Re: Pairing problem

Just for fun, can you write the two numbers assigned to person number one, the two numbers assigned to person number two, then add each pair of numbers...I think that might move things in the right direction

5. ## Re: Pairing problem

So the proof is the pairs must add to 76?? That seems rather bland..?

6. ## Re: Pairing problem

Hello, mjoshua!

Each of the 75 children in a line was assigned one of the integers from 1 through 75 by counting off in order.
Then, standing in the same order, the children counted off in the opposite direction,
so that the child who was assigned the number 75 the first time was assigned the number 1 the second time.
Which of the following is a pair of numbers assigned to the same child?

How can I show and prove that it has to be 47 and 29?

Take a look at GJA's observation . . .

. . $\begin{array}{c|cccccc} \text{1st count} & 1 & 2 & 3 & \cdots & 74 &75 \\ \hline \text{2nd count} & 75 & 74 & 73 & \cdots & 2 & 1 \end{array}$

Each child received a pair of numbers whose sum is 76.

Yes, that's kind of bland.
But when an answer is that simple, don't expect fireworks . . .